Question

which equation can be used to find the length of \\( \overline { a c } \\)?

\\( ( 10 ) \sin \left( 40 ^ { \circ } \
ight) = a c \\)
\\( 10 \cos \left( 40 ^ { \circ } \
ight) = a c \\)
\\( \frac { 10 } { \sin \left( 40 ^ { \circ } \
ight) } = a c \\)
\\( \frac { 10 } { \cos \left( 40 ^ { \circ } \
ight) } = a c \\)

Explanation:

Step1: Recall the sine function definition

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\).

Step2: Identify the sides and angle

Here, \(\theta = 40^{\circ}\), the hypotenuse \(AB = 10\) in, and the side opposite to \(\angle B\) is \(AC\).

Step3: Apply the sine formula

By \(\sin B=\frac{AC}{AB}\), substituting \(B = 40^{\circ}\) and \(AB = 10\), we get \(\sin(40^{\circ})=\frac{AC}{10}\), which can be rewritten as \((10)\sin(40^{\circ})=AC\).

Answer:

\((10)\sin(40^{\circ}) = AC\)